Metric expansion of space

The metric expansion of space is a key part of science's current understanding of the universe, whereby space itself is described by a metric which changes over time. It explains how the universe expands in the Big Bang model which all cosmological experiments and measurements to date support.

The expansion of space in this way is conceptually different from other kinds of expansions and explosions that are seen in nature. Current relativistic theories of spacetime suggest that what we see as "space" and "distance" are not absolutes, but are determined by a metric that can change. Rather than objects in a fixed "space" moving apart into "emptiness", it is the metric of the space those objects are in, which is itself changing.

To an observer, it is as if without objects themselves moving, space is somehow "growing" in between them. Because it is the metric defining distance which is changing, rather than objects moving in space, this growth (and the resultant movement apart of non-bound objects) is not restricted by the speed of light upper bound that results from special relativity.

Theory and observations suggest that very early in the history of the universe, there was an "inflationary" phase where this metric changed very rapidly, and that the remaining time-dependence of this metric is what we observe as the so-called Hubble expansion, the moving apart of all gravitationally unbound objects in the universe. The expanding universe would therefore be a fundamental feature of the spacetime we inhabit. This expansion describes a universe fundamentally different from the static universe Albert Einstein first considered when he developed his gravitational theory.

Overview

A metric defines how a distance can be measured between two points in space. In Euclidean geometry, this distance can be measured by tracking a straight line between two points. However, in non-Euclidean geometry, the notion of a "straight line" or "distance" may not be the same, and the notion of "distance" varies depending upon the actual metric involved. In this sense, a metric is a generalization of the concept of "distance" for other geometries, much as "negative numbers" and "complex numbers" are a generalization of the everyday concept of the usual natural numbers (1, 2, 3, etc). According to current observations, space appears to have a Euclidean geometry, but this may be an illusion: On a larger scale, space may in fact be non-Euclidean while the notion of "distance" and space on such a scale would be described by a metric that does not behave as expected from "flat" geometry.

Technically, the metric expansion of space is a feature of many solutions to the Einstein field equations of general relativity. In particular, if the cosmological principle is assumed with a time-varying universe the simplest solution allows for the proper distances in space to change with an evolving scale factor. This theoretical explanation provides a clean explanation of the observed Hubble Law which indicates that galaxies that are more distant from us appear to be receding faster than galaxies that are closer to us. In spaces that expand, the metric changes with time in a way that causes distances to appear larger at later times, so in our Big Bang universe, we observe phenomena associated with metric expansion of space. If we lived in a space that contracted (a Big Crunch universe) we would observe phenomena associated with a metric contraction of space instead.

Early theoretical models predicted that a universe which was dynamical and contained ordinary gravitational matter would contract rather than expand. Einstein addressed this problem by adding a cosmological constant into his theories to balance out the contraction, in order to obtain a static universe solution. It wasn't until the observations of Edwin Hubble confirmed a metric expansion of the universe that scientists accepted that the universe was expanding. Until the 1980s, no one had an explanation for why this was the case, but with the development of models of cosmic inflation, the expansion of the universe became a general feature resulting from vacuum decay. Accordingly, the question "why is the universe expanding?" is now answered by understanding the details of the inflaton decay process which occurred in the first 10^-32 seconds of the existence of our universe. It is suggested that in this time the metric changed exponentially, causing space to change from smaller than an atom to around 100 million light years across.

Measuring distances in expanding spaces

In expanding space, proper distances are dynamical quantities which change with time. An easy way to correct for this is to use comoving coordinates which remove this feature and allow for a characterization of the universe as a whole without having to characterize the physics associated with metric expansion. In comoving coordinates, the distances between all objects are fixed and the instantaneous dynamics of matter and light are determined by the normal physics of gravity and electromagnetic radiation. Any time-evolution however must be accounted for by taking into account the Hubble law expansion in the appropriate equations. Cosmological simulations that run through significant fractions of the universe's history therefore must be able to work in physical units which can directly predict observational cosmology.

An easy way to take into account the expansion of the universe is to integrate through conformal time which is a time-like geodesic that is scaled by the scale factor. If the functional form of the scale factor is known from the Friedman equations and two events are decided upon, the dynamics associated with the expanding universe between those events are summarized by such a mathematical manipulation.

Model analogies

Because metric expansion is not seen on the physical scale of humans, the concept may be difficult to grasp. Two analogies, the ant-on-a-balloon analogy and the raisin bread analogy, have been developed to aid in conceptual understanding. Each analogy has its benefits and drawbacks.

Ant on a balloon model

The ant on a balloon model is a two-dimensional analog for three-dimensional metric expansion. An ant is imagined to be constrained to move on the surface of a huge balloon which to the ant's understanding is the total extent of space (see article on flatland for more consequences of a two-dimensional constraint). At an early stage of the balloon-universe, the ant measures distances between separate points on the balloon which serves as a standard by which the scale factor can be measured. The balloon is inflated some more, and then the distance between the same points is measured and determined to be larger by a proportional factor. The surface of the balloon still appears flat, and yet all the points have appeared to recede from the ant, indeed every point on the surface of the balloon is proportionally farther from the ant than earlier in the life of the balloon universe. This explains how an expanding universe can result in all points receding from each other simultaneously. No points are seen to get closer together.

In the limit where the ant is tiny and the balloon is enormous, the ant also cannot detect any curvature associated with the geometry of the surface (which is roughly an elliptical geometry for the outside surface of a curving balloon). To the ant, the balloon appears to be a plane extending out in all directions. This mimics the so-called "flatness" seen in our own observable universe which appears even on the largest to follow the geometrical laws associated with flat geometry. Like the ant on an enormous balloon, while we may be unable to detect curvature, on larger, unobservable scales there may be residual curvature. The shape of the universe we observe is driven to be flat no matter what starting conditions the universe had by the same cosmic inflation which caused the universe to begin expanding in the first place.

In the analogy, the two dimensions of the balloon do not expand "into" anything since the surface of the balloon admits infinite paths in all directions at all times. There is some possilibity for confusion in this analogy since the balloon can be seen by an external observe to be expanding "into" the third dimension (in the radial direction), but this is not a feature of metric expansion, rather it is the result of the arbitrary choice of the balloon which happens to be a manifold embedded in a third dimension. This third dimension is not mathematically necessary for two-dimensional metric expansion to occur, and the ant that is confined to the surface of the balloon has no way of determining whether a third dimension exists or not. It may be useful to visualize a third dimension, but the fact of expansion does not theoretically require such a dimension to exist. This is why the question "what is the universe expanding into?" is poorly phrased. Metric expansion does not have to proceed "into" anything. The universe that we inhabit does expand and distances get larger, but that does not mean that there is a larger space into which it is expanding.

Raisin bread model

The raisin bread model imagines galaxies as raisins in a raisin bread dough that will "rise" or "expand" when cooked. As the expansion occurs, each of the raisins gets farther from each of the other raisins while the raisins themselves stay the same size. The dough between raisins in this model acts as the space between galaxies while the raisins as "bound objects" are not subject to the expansion. This model is useful for explaining how it is that a standard ruler can be determined for measuring the expansion. In an empty universe, space serves as the only ruler and as rulers expand with space, there would be no way to distinguish between an expanding universe and a static universe. Only in a universe where there are objects which are bound and do not expand so that the rulers are independent of the expansion can the metric expansion be measured.

Like the ant on the balloon model, this model also suffers from the problem that the raisin bread is expanding into the pan. To make the analogy to the universe, it is necessary to imagine a raisin bread that has no observable edge. Expansion would still occur, but the question "what is the raisin bread expanding into?" would be meaningless.

References

External links

Physical cosmology | General relativity